🚀 go-pugleaf

>
>A waterfall display of FFTs draws every new FFT curve on the screen
>slightly below and "in front" of the last curve.  The effect is that
From: harrison@sunny.DAB.GE.COM (Gregory Harrison)
Path: sunny!harrison

Another type of transform can be used to obtain the time-frequency 
display of a waterfall display.  This is the Wigner Distribution.
The Wigner Distribution performs an operation on the input time series, then 
does an FFT.  The input time series is sampled starting at progressive
times in the data, as in the FFT method.  What the Wigner Distribution, WD
does is to provide greater time-frequency localization of the components
in  the signal.  

In a FFT, as the FFT length is increased, the frequency resolution 
is increased, but the ability to discern changing frequencies decreases.
As series of FFTs are displayed on the waterfall, the ability to 
discern the time of presence of a signal also decreases (I think 
this is also a direct function of the length).  This results in 
signals being smeared in time and frequency, if they are nonstationary.

The Wigner calculates the Power spectrum at a certain time, not over a  

range of time.  The length of the WD can be increased arbitrarily
in order to obtain frequency resolution, without introducing 
smearing.

For instance, the WD of a chirp signal is very close to a nice clear
delta function that increases in frequency as time goes on.  The
FFT technique yields a messy, smeared, wide band of frequencies
increasing in frequency as time goes on.  

The WD introduces artifacts not in the FFT, but a large number of
these artifacts can be eliminated using the Wigner-Ville Distribution,
WVD.  The Wigner provides better high frequency response than the
FFT, as seen when analyzing speech with identical length FFTs and 
WDs.

The Wigner operates by creating a kernal:

k(i) = s(i)s*(N-i)

for i = 1 to N,

k(i) is the kernal, and s(i) is the input sequence.

Then the following:

WD(f) = abs(FFT(k(i)))

There are ways to make the WD provide real output for real input, and thus
perform 2 WD in one FFT essentially doubling throughput.  A very good
reference is:

B. Boashash and P.J. Black, "An Efficient Real-Time Implementation
of the Wigner-Ville Distribution," IEEE Trans. Acoustics, Speech and
Signal Processing, Vol. ASSP-35. No. 11, Nov. 1987

P.S. This is what I'm doing my thesis on.

Greg Harrison
My opinions are not intended to reflect those of GE.